The fractional Poisson process and the inverse stable subordinator

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.


Introduction
The fractional Poisson process (FPP) was introduced and studied by Repin and Saichev [42], Jumarie [22], Laskin [27], Mainardi et al. [30,32], Uchaikin et al. [47] and Beghin and Orsingher [5,6]. The FPP is a natural generalization of the usual Poisson process, with an interesting connection to fractional calculus. This renewal process has IID waiting times J n that satisfy (1.1) P(J n > t) = E β (−λt β ) for 0 < β ≤ 1, where denotes the Mittag-Leffler function. When β = 1, the waiting times are exponential with rate λ, since e z = E 1 (z). Let T n = J 1 + · · · + J n be the time of the nth jump. Then the FPP is a renewal process with Mittag-Leffler waiting times. A compound FPP is obtained by subordinating a random walk to the FPP. The resulting process is non-Markovian (unless β = 1) and the distribution of that process solves a "master equation" analogous to the Kolmogorov equation for Markov processes, with the usual integer order time derivative replaced by a fractional derivative.
The research of MMM was partially supported by NSF grants DMS-0803360 and EAR-0823965. 1 The continuous time random walk (CTRW) is another useful model in fractional calculus. Consider a CTRW whose IID particle jumps Y n have PDF w(x), and whose IID waiting times (J n ) are Mittag-Leffler variables independent of (Y n ). The particle location after n jumps is S(n) = Y 1 + · · · + Y n , and the CTRW S(N β (t)) gives the particle location at time t ≥ 0. Hilfer and Anton [19] show that the PDF p(x, t) of the CTRW S(N β (t)) solves the fractional master equation where ∂ β t denotes the Caputo fractional derivative. The Caputo fractional derivative, defined for 0 ≤ n − 1 < β < n by where g (k) denotes the k-th derivative of g, was invented to properly handle initial values [11].
If β = 1, then ∂ β t is the usual first derivative. The corresponding CTRW S(N 1 (t)) is a compound Poisson process, and (1.4) reduces to the Cauchy problem associated with this infinitely divisible Lévy process. Then a general result on Cauchy problems [2,Theorem 3.1] implies that the PDF of the time-changed process S(N 1 (E(t))) solves the fractional Cauchy problem (1.4), where (1.7) E(t) = inf{r > 0 : D(r) > t} is the right-continuous inverse (hitting time, first passage time) of D(t), a standard β-stable subordinator with E[e −sD(t) ] = e −ts β for some 0 < β < 1.
Since the PDF of both S(N β (t)) and S(N 1 (E(t))) solve the same governing equation (1.4), with the same point-source initial condition (i.e., both processes start at the origin), these two processes have the same one dimensional distributions. Heuristically, the degenerate case Y n ≡ 1 gives S(n) = n, which strongly suggest that the FPP N β (t) and the process N 1 (E(t)) have the same one dimensional distributions. We will call N 1 (E(t)) the fractal time Poisson process (FTPP), since it comes from a self-similar time change (see, e.g., [34,Proposition 3.1]). In this paper, we will prove that the FPP and the FTPP are in fact the same process, by showing that the waiting times between jumps in the FTPP are IID Mittag-Leffler. This strong connection between the FPP and the FTPP unifies the two main approaches in the stochastic theory of fractional diffusion. For example, the FPP approach was used recently in the work of Behgin and Orsingher [5], while the inverse stable subordinator is a key ingredient in [38].

Two equivalent formulations
Recall that the fractional Poisson process (FPP) N β (t) is a renewal process with Mittag-Leffler waiting times (1.1), and the fractal time Poisson process (FTPP) N 1 (E(t)) is Poisson process, with rate λ > 0, time-changed via the inverse stable subordinator (1.7). The proof that the FPP and the FTPP are the same process requires the following simple lemma.
Lemma 2.1. Let D(t) be a strictly increasing right-continuous process with left-hand limits, and let E(t) be its right-continuous inverse defined by (1.7). Then Proof. Let t 0 = sup{t > 0 : E(t) < r}. Then there exists a sequence of points t n ↑ t 0 such that E(t n ) < r for all n.
Since D has left-hand limits, for any r n ↑ r we have D(r n ) → D(r−) as n → ∞. If D(r−) > t 0 , then for some r n < r we have D(r n ) > t 0 . Since E(t) is nondecreasing and continuous, this implies that E(D(r n )) ≥ r, by definition of t 0 . But, E(D(r)) = r for all r > 0 implying that r n ≥ r, which is a contradiction. Thus, (2.1) follows.
Theorem 2.2. For any 0 < β < 1, the FTPP N 1 (E(t)) is also a FPP. That is, the waiting times between jumps of the FTPP are IID Mittag-Leffler.
Proof. Let W n be an IID sequence with P(W n > t) = e −λt and V n = W 1 + · · · + W n so that the Poisson process N 1 (t) = max{n ≥ 0 : V n ≤ t}. Let (2.2) τ n = sup{t > 0 : N 1 (E(t)) < n} denote the jump times of the FTPP. This definition of the jump times takes into account the fact that E(t) has constant intervals corresponding to the jumps of the process D(t). Using the fact that {N 1 (t) < n} = {V n > t} for the Poisson process, along with (2.2), we have Then Lemma 2.1 implies that τ n = D(V n −). Define X 1 = τ 1 and X n = τ n − τ n−1 for n ≥ 2, the waiting times between jumps of the FTPP. In order to show that the FTPP is an FPP, it suffices to show that X n are IID Mittag-Leffler, i.e., they are IID with J n . Recall that the Laplace transform of the exponential distribution E(e −sWn ) = λ/(λ + s). Also recall that E(e −sD(t) ) = e −ts β . Since D(t) is a Lévy process, it has no fixed points of discontinuity and hence D(t−), D(t) are identically distributed for all t ≥ 0. Then a conditioning argument yields see for example [38,Eq. (3.4)]. Now integrate by parts to see that and then the uniqueness theorem for LT implies that T 1 , τ 1 are identically distributed. In particular, X 1 has the same Mittag-Leffler distribution as J 1 .
A straightforward extension of this argument shows that (T 1 , . . . , T n ) is identically distributed with (τ 1 , . . . , τ n ) for any positive integer n. To ease notation, we only write the case n = 2. First observe that using the independence of J 1 and J 2 . Next write using the fact that D(t) has independent increments. Then Now an application of the continuous mapping theorem shows that (J 1 , . . . , J n ) is identically distributed with (X 1 , . . . , X n ) for any positive integer n. Then (X n ) is an IID sequence, so N 1 (E(t)) is a renewal process. Remark 2.3. Theorem 2.2 extends a result in Behgin and Orsingher [5]. They define (in our notation) a random variable E(t) and show that the two random variables N β (t) and N 1 (E(t)) have the same density function, by comparing their Laplace transforms. They identify E(t) only though its density function, which they express in terms of an integral involving the density of D(t), see Remark 3.2 for more detail. Cahoy, Uchaikin, Woyczynski [10] also connect the Mittag-Leffler distribution with a stable law. They note that (in our notation) P (J n > t) = E[exp(−λt β /D(1) β )], which is useful in simulations. To connect this with our work, note that E(t) = (t/D(1)) β in distribution (see Corollary 3.1 in [34]), so that P (J n > t) = E[exp(−λE(t))]. A result of Bingham [8] shows that the the Laplace transform of the stable hitting time E(t) is Mittag-Leffler, so that (1.1) holds.
Remark 2.4. The proof of Theorem 2.2 uses the fact that, if D(t) is a β-stable subordinator and W 1 is exponential, then D(W 1 ) has a Mittag-Leffler distribution. This fact was first noticed by Pillai [40], who showed that W 1/β 1 D(1) is Mittag-Leffler. These are equivalent because D(t) is identically distributed with t 1/β D(1). This Mittag-Leffler distribution is also known as the positive Linnik law, e.g., see Huillet [21]. It has the property of geometric stability: A geometric random sum of Mittag-Leffler variables is again Mittag-Leffler, e.g., see Kozubowski [25].
Next we want to show that the FTPP N 1 (E(t)), and hence also the FPP N β (t), occurs naturally as a CTRW scaling limit. This provides a further justification for the FPP as a robust physical model, see for example Laskin [27]. Suppose now that P(J n > t) = t −β L(t), where 0 < β < 1 and L is slowly varying. For example, this is true of the Mittag-Leffler waiting times. Then J 1 belongs to the strict domain of attraction of some stable law D with index 0 < β < 1, i.e., there exist b n > 0 such that where D(1) = D > 0 almost surely, and ⇒ denotes convergence in distribution. Let . Then b(t) = t −1/β L 0 (t) for some slowly varying function L 0 (t) (e.g., see [17,XVII.5]). Since b varies regularly with index −1/β, b −1 is regularly varying with index 1/β > 0 and so by [46, Property 1.5.5] there exists a regularly varying functioñ b with index β such that 1/b(b(c)) ∼ c, as c → ∞. Here we use the notation f ∼ g for positive functions f, g if and only if f (c)/g(c) → 1 as c → ∞. Let T n = J 1 + · · · + J n and define a renewal process with these waiting times. Next, construct a CTRW with iid Bernoulli jumps Y n , a binomial random variable. Then S (p) (R(t)) is a CTRW with heavy tailed waiting times and Bernoulli jumps.
Theorem 2.5. The FTPP is the process limit of a CTRW sequence: Proof. Since the sequence (J n ) is in the strict domain of attraction of a β-stable random variable D, [34,Corollary 3.4] shows that as p → 0, using the fact that (1 + ap) 1/p → e a as p → 0. It follows by the continuity theorem for LT that Then a standard argument (e.g., see [33,Example 11.2.18] shows that we also get =⇒ denotes convergence of all finite dimensional distributions. Since the sample paths of S (p) ([λt/p]) are increasing and N 1 (t) is continuous in probability, being a Lévy process, J 1 convergence follows using [8,Theorem 3].
Since the CTRW waiting times (J n ) are independent of the jumps (Y (p) n ), and since 1/b(c) → 0 as c → ∞, it follows that Remark 2.6. For the specific case of Mittag-Leffler waiting times, where P(J n > t) = E β (−t β ), we can take b n = n −1/β in (2.5). To check this, note that

Fractional calculus
This section develops some interesting connections between the fractional Poisson process and fractional calculus. In the process, some apparent inconsistencies in the existing literature will be explained. Behgin and Orsingher [5,Eq. (2.17)] show that the FPP of order 0 < β < 1 has distribution where T t is a random process with PDF V (x, t) for t > 0. However, that process is identified only in terms of its one dimensional distributions (PDF). Theorem 2.2 shows that the inverse stable subordinator E(t) is one such process.
On the other hand, a simple conditioning argument shows that the equivalent FTPP process has distribution where h(x, t) is the density of E(t), a PDF on x > 0 for each t > 0. It follows from [36, Theorem 4.1] that this PDF solves . In view of Theorem 2.2, the two distributions (3.1) and (3.3) must be equal. Thus, the main purpose of this section is to reconcile the two fractional differential equations (3.2) and (3.4).
In particular, the two fractional partial differential equations (3.2) and (3.4) are consistent, in the sense that the folded solution V ( Proof. Mainardi [29,Eq. (3.2)] shows that the solution to the fractional diffusion-wave equation (3.2) has LT Since both are differentiable in t, they are also continuous, so LT uniqueness for continuous functions implies (3.5). Take Fourier transforms in (3.6) to see that the solution to (3.2) has Fourier-Laplace transform (FLT) where we have used the fact that e −a|x| has FT 2a/(a 2 + k 2 ). Rearrange to get and invert the FT to get This is easy to verify from the definition (1.5), using the corresponding formula for the integer derivative, along with the fact that s β−1 is the LT of t −β /Γ(1 − β). Now use the remaining initial condition ∂ t v(x, 0) ≡ 0 for 1/2 < β < 1 to invert the LT, and arrive at (3.2).
The equivalence in Theorem 3.1 results from folding the solution to the fractional diffusion-wave equation (3.2). Another fractional partial differential equation for the density h(x, t) of the standard inverse β-stable subordinator E(t), which is closer to the form (3.2), can be obtained by arguments similar to those used in [3] to connect the inverse stable subordinator to iterated Brownian motion. In that theory, it is customary to avoid distributions by imposing a functional initial condition.
Theorem 3.4. Let E(t) be the standard inverse β-stable subordinator with density h(x, t). Then for any f ∈ L 2 (R) ∩ C 1 (R), the function solves the fractional differential equation In particular, when β = 1/2, (3.12) solves and in this case we also have u( Proof. From (3.9), we havē Then E(t) and |B(t)| have the same one dimensional distributions, so we also have u(x, t) = E x [f (|B(t)|)]. Note that X(t) = X(0) − t is a fortiori a continuous Markov process associated with the shift semigroup
for t > 0 and x ∈ R, which is then equivalent to (3.4). The proof is similar to Theorem 3.4. For example, when β = 1/3 usē

Renewal processes and inverse subordinators
Theorem 2.2 shows that a Poisson process, time-changed by an inverse stable subordinator, yields a renewal process with Mittag-Leffler waiting times. This section extends that result to arbitrary subordinators that are strictly increasing. Let D(t) be a strictly increasing Lévy process (subordinator) with E[e −sD(t) ] = e −tψ D (s) , where the Laplace exponent   E(t)) is a renewal process whose IID waiting times (J n ) satisfy Proof. The proof is similar to Theorem 2.2. Take N 1 (t) = max{n ≥ 0 : V n ≤ t}, where V n = W 1 + · · · + W n , with W n IID as P(W n > t) = e −λt . Let τ n = sup{t > 0 : N 1 (E(t)) < n} = sup{t > 0 : E(t) < V n } and apply Lemma 2.1 to get τ n = D(V n −). Then, as in the proof of Theorem 2.2, we have . .
Integrate by parts to get (4.6) and condition to get .
On the other hand, using the fact that (J n ) are IID. To finish the proof, use continuous mapping to show that (J 1 , . . . , J n ) is identically distributed with (X 1 , . . . , X n ), where X n = τ n − τ n−1 are the waiting times between jumps for the process N 1 (E(t)). where T n = n i=1 J i and (J n ) are IID according to (4.3). Theorem 4.1 shows that N D (t) = N 1 (E(t)). This extends the relation N β (t) = N 1 (E(t)) from Theorem 2.2, the special case of an inverse stable subordinator E(t) and Mittag-Leffler waiting times J n , to a general inverse subordinator.
where h(x, t) is the density of E(t). A straightforward extension of the argument in Remark 3.3 shows that which reduces to (3.10) in the special case ψ D (s) = s β for a stable subordinator D(t). Use (4.5) and (4.6) to see that the first factor in (4.8) is the LT (t → s) of h(λ, t) = P(J n > t) = E(e −λE(t) ), and the second is the LT of T n = J 1 + J 2 + · · · + J n with J n IID as in (4.3). Denote the distribution of T n by F (n, * ) , the n-fold convolution of the distribution function F of J 1 . Invert the LT to get (4.9) p D (n, t) = t 0h (λ, t − s)F (n, * ) (ds) which extends (3.11).

CTRW scaling limits and governing equations
In this section, we extend the fractional calculus results of Section 3 to the inverse subordinators of Section 4. A general theory of CTRW scaling limits and governing equations is developed in [36]. Consider a sequence of CTRW indexed by a scale parameter c > 0. Take J c n nonnegative IID random variables representing the waiting times between particle jumps and T c (n) = n i=1 J c i , the time of the nth jump. Let Y c i be IID random vectors on R d representing the particle jumps, independent of the waiting times, and set S c (n) = n i=1 Y c i , the location of the particle after n jumps. Define N c t = max{n ≥ 0 : T c (n) ≤ t}, the number of jumps by time t ≥ 0 and Y c i 13 the position of the particle at time t ≥ 0 and scale c > 0. Assume a triangular array limit in the J 1 topology on D([0, ∞), R d × R + ), so that A(t) and D(t) are independent Lévy processes on R d and R, respectively. Since the waiting times are nonnegative, D(t) is a subordinator. In this section, we assume the drift b = 0 in (4.1), as well as condition (4.2) and Assumption (4.2) implies that the process {D(t)} is strictly increasing, i.e., D(t) is not compound Poisson. Then [36,Theorem 3.1] shows that the inverse subordinator E(t) in (1.7) has a Lebesgue density 3) hold. The distribution function of the CTRW limit process A(E(t)) in (5.5) is given by where h(u, t) is the density (5.4) of the inverse subordinator E(t). The distribution function Q(x, t) solves the generalized Cauchy problem in the mild sense, where H(x) = I(x ≥ 0) is the Heaviside function. Furthermore, P (x, u) solves the Cauchy problem and h(x, t) solves the inhomogeneous Cauchy problem Proof. The proof is similar to [36,Theorem 4.1]. Equation (5.6) follows from a simple conditioning argument. Apply [36,Theorem 3.6] to see that Q(x, t) has FLT and rearrange to get From [36, Eq. (3.12)] we get Now invert the FLT (5.11), using (5.12) and e −ik·x H(dx) ≡ 1, to arrive at (5.7). It is well known that P (x, t) solves the Cauchy problem (5.8), see for example [20]. Equation This rearranges to λh(ξ, s) = −ψ D (s)h(λ, s) + s −1 ψ D (s). Inverting the LLT using (5.12) to see that h(x, t) solves (5.9).
For any random walk S(n) = n i=1 Y i , the compound Poisson process A(t) = S(N 1 (t)) is a Lévy process. Introduce IID waiting times (4.3) between these random walk jumps to get a CTRW. In this case, the CTRW is exactly of the form A(E(t)), without passing to the limit. Then the governing equations in Theorem 5.1 pertain to the CTRW itself.
Theorem 5.2. Assume D(t) is a subordinator without drift such that conditions (4.2) and (5.3) hold, and let E(t) be the inverse subordinator (1.7). Take J n IID waiting times according to (4.3), and let N D (t) denote the renewal process (4.7). Take Y n IID jumps on R d , independent from (J n ), with common distribution µ, and let S(n) = n i=1 Y i . Then the distribution function P (x, t) = P(X(t) ≤ x) of the CTRW X(t) = S(N D (t)) solves the generalized Cauchy problem in the mild sense. Furthermore, X(t) = A(E(t)), where A(t) = S(N 1 (t)) is a compound Poisson process.
A standard conditioning argument shows that the compound Poisson FTP (k, t) = e −tψ A (k) , where the Fourier symbol ψ A (k) = λ(1−μ(k)). The inverse FT of ψ A (k)f (k) is using the FT convolution property. Now Theorem 5.1 implies that (5.13) holds.
This is the Kolmogorov forward equation for the Markov process A(t). If µ has density w(x), apply ∂ x on both sides of (5.15) to see that the probability density p(x, t) = ∂ x P (x, t) of A(t) solves (1.6). If D is the stable subordinator with Laplace symbol ψ D (s) = s β , then (5.13) holds with φ D (t, ∞) = t −β /Γ(1 − β) and ψ D (∂ t ) = D β t , the Riemann-Liouville fractional derivative. The Riemann-Liouville fractional derivative is defined for 0 ≤ n − 1 < β < n by which differs from the Caputo derivative (1.5) in that the derivative is applied after the integration. The LT of D β t g(t) is s βg (s). Apply ∂ x to both sides of (5.13) in this case to get the fractional kinetic equation of Zaslavsky [49]. To recover (1.4), use ∂ β t g(t) = D β t g(t) − g(0)t −β /Γ(1 − β) and p(x, 0) = δ(x). Remark 5.4. In the special case where µ = ε 1 is a point mass, so that Y n = 1 almost surely, A(t) = N 1 (t) is a Poisson process with rate λ > 0. Then the distribution function P (x, t) of the renewal process N D (t) = A(E(t)) solves If D is the stable subordinator with Laplace symbol ψ D (s) = s β , Equation (5.17) reduces to Remark 5.3. The probability mass function p(n, t) = P (n, 1) − P (n − 1, t) = ∆P (n, t) for n > 0. Apply the difference operator ∆ on both sides to obtain ∂ β t p(n, t) = −λ[p(n, t) − p(n − 1, t)] as in Jumarie [22].
Remark 5.5. Scher and Lax [45] showed that a CTRW with waiting time distribution ω and jump distribution ν has FLT , and then it follows that ψ D (s) = λ(1 −ω(s))/ω(s). The jumps Y n in Theorem 5.2 have Fourier symbol ψ A (k) = λ(1 −μ(k)) and then (5.10) implies which provides a different proof that the CTRW equals A(E(t)) in this case. To simulate the sample paths of the non-Markovian process A(E(t)), it is sufficient to simulate the CTRW. In particular, the renewal process N D (t) gives the exact jump times of the inverse subordinator E(t).
Remark 5.6. In the general case, where A(t) is not compound Poisson, Theorem 5.2 provides a useful approximation. Given a Lévy process A(t), take Y n = A(n) − A(n − 1), so that S(n) = A(n). Take N(t) a Poisson process with rate 1, so that S(λ −1 N(λt)) is compound Poisson with Fourier symbol Then S(λ −1 N(λt)) ⇒ A(t) as λ → ∞, and the CTRW with IID waiting times (4.3) and these compound Poisson jumps converges to A(E(t)) as λ → ∞. As in Remark 5.5, this fact can be used to simulate sample paths of the process A(E(t)). This fact has been exploited by Fulger, Scalas and Germano [18] to develop fast simulation methods for space-time fractional diffusion equations.
Example 5.7. Tempered stable subordinators are theoretically interesting [4,43] and practically useful [16,37]. Take D(t) tempered stable with Laplace symbol ψ D (s) = (s + a) β − a β for a > 0 and 0 < β < 1, and let E(t) be its inverse (1.7). Theorem 4.1 shows that N 1 (E(t)) is a renewal process. Let (τ n ) denote the arrival times of this renewal process, and use (4.4) to get This tempered fractional Poisson process N 1 (E(t)) has tempered Mittag-Leffler waiting times, but with a different rate parameter: Use (2.4) to see that the Mittag-Leffler PDF f (t) = ∂ t [1 − E β (−ηt β )] has Laplace transform η/(η + s β ), and so Of course f (t)e −at is not a PDF, and in fact we have (set s = 0 above) Then the tempered Mittag-Leffler PDF f a (t) = f (t)e −at (η + a β )/η has LT ∞ 0 e −st f a (t)dt = η + a β η + (s + a) β = λ λ + (s + a) β − a β = E(e −sτ 1 ) when η + a β = λ. Cartea and Del-Castillo [12] show that the tempered fractional derivative ψ D (∂ t )g(t) = e −at ∂ β t [e at g(t)]−a β g(t). It is also known (e.g., see [4]) that the corresponding Lévy measure is exponentially tempered: ψ D (dt) = e −at ψ(dt), where ψ(t, ∞) = t −β /Γ(1 − β) is the Lévy measure of the standard β-stable subordinator. Then Theorem 5.2 shows that the CTRW with tempered Mittag-Leffler waiting times and compound Poisson jumps solves a tempered fractional Cauchy problem given by (5.14) and φ D (t, ∞) = β ∞ t e −at t −β−1 dt/Γ(1 − β). More generally, Theorem 5.1 shows that the distribution function of the CTRW scaling limit A(E(t)) is governed by this equation, with the corresponding operator ψ A (−iD x ). Apply ∂ x on both sides of (5.17) to see that the PDF of the renewal process with tempered Mittag-Leffler waiting times solves e −at ∂ β t [e at p(x, t)] − a β p(x, t) = −λ[p(x, t) − p(x − 1, t)] + δ(x)φ D (t, ∞). A wide variety of tempered stable models in R d are discussed in Rosiński [43]. Random walks in R d with tempered stable scaling limit are developed in [13]. For exponentially tempered stable waiting times in R 1 , a renewal process with tempered Mittag-Leffler waiting times gives the same process exactly, without taking limits. This can be useful for simulating sample paths.
Example 5.8. Chechkin et al. [15,14] used distributed order fractional derivatives to model multi-scale anomalous subdiffusion, where a different power law pertains at short and long time scales, and ultraslow diffusion, for a plume of particles spreading at a logarithmic rate. Given a finite Borel measure ν on (0, 1), the distributed order fractional derivative is defined by where ∂ β t is the Caputo fractional derivative (1.5). If ν is discrete, this is a linear combination of fractional derivatives. Let D(t) be the distributed order stable subordinator with Laplace symbol ψ D (s) = s β ν(dβ) and E(t) its inverse (1.7). Let ν(dβ) = p(β)dβ for some p ∈ C 1 (0, 1), then by (2.19) in Kochubei [24] (5. 19) P (J n > t) = E(e λE(t) ) = λ π Substitute (5.19) into (4.9) to obtain an explicit formula for the probability mass function of the distributed order Poisson process.